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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Subhajit Bandyopadhyay

Subhajit Bandyopadhyay's wiki page.

Subhajit Bandyopadhyay has authored 3 sequences.

A373535 Number of self-conjugate 10 core partitions of n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 2, 3, 8, 4, 6, 7, 7, 9, 6, 7, 8, 10, 8, 10, 11, 9, 8, 10, 12, 11, 16, 14, 12, 17, 13, 13, 17, 13, 17, 18, 18, 17, 17, 15, 18, 19, 17, 18, 18, 17, 24, 28, 16, 25, 26, 24, 22, 25, 26, 25, 28, 22, 31, 31, 25, 33, 39, 32, 31, 37
Offset: 0

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Author

Subhajit Bandyopadhyay, Aug 23 2024

Keywords

Comments

The generating function for sc_t(n), the number of t-core partitions of n, is given in Section 7 of the article titled 'Cranks and t-cores' by Garvan, Kim, and Stanton (1990). This sequence is sc_10(n).

Links

  • Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and t-cores, Invent. Math. 101 (1990):1-17.

Crossrefs

Cf. A182805.

Programs

  • PARI
    q='q+O('q^80);
chi(q) = eta(q^2)^2 / (eta(q)*eta(q^4));
gf = chi(q) * eta(q^20)^5;
Vec(%)

Formula

G.f.: chi(q) * f(-q^20)^5, where chi(q) is Ramanujan's chi function (see A000700) and f(q) is Ramanujan's theta function f(q) (see A121373).

A368495 Psi-analog of the 5-core partition function.

Original entry on oeis.org

1, 1, 1, 2, 3, -1, 0, 2, 0, -2, 6, 6, 3, 5, 8, 0, 0, 1, 0, 0, 11, 6, 6, 12, 10, -6, 0, 6, 0, -5, 12, 16, 9, 7, 18, 0, 0, 3, 0, 0, 21, 12, 7, 22, 23, -6, 0, 12, 0, -12, 26, 20, 12, 18, 20, 0, 0, -2, 0, 0, 21, 21, 18, 24, 33, -16, 0, 18, 0, -7, 36, 36, 13, 20, 36, 0, 0, 9, 0, 0, 41, 12, 24, 42, 30, -12, 0, 14, 0, -22
Offset: 0

Views

Author

Subhajit Bandyopadhyay, Dec 27 2023

Keywords

Comments

a(n) is a natural analog to A368490 and the 5-core partition function (see A053723). They have similar generating functions.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122) and psi(q) (A010054)

Links

Crossrefs

Cf. A053723, A368490.

Programs

  • PARI
    q='q+O('q^99);
rpsi(q)=eta(q^2)^2 / eta(q);
gf=rpsi(-q^5)^5/rpsi(-q);
Vec(%) \\ Joerg Arndt, Dec 27 2023

Formula

G.f.: psi(-q^5)^5/psi(-q), where psi(q) is the Ramanujan's theta function psi (see A010054)

A368490 Phi-analog of the 5-core partition function.

Original entry on oeis.org

1, 2, 4, 8, 14, 14, 20, 24, 20, 14, 32, 24, 24, 48, 60, 32, 62, 64, 28, 40, 66, 24, 48, 88, 68, 74, 120, 80, 72, 60, 76, 64, 84, 96, 160, 96, 98, 144, 80, 48, 128, 84, 48, 168, 168, 98, 220, 184, 88, 86, 172, 64, 144, 208, 200, 168, 204, 160, 120, 120, 128, 124, 128, 168, 254, 192, 240, 264, 192, 88, 228, 144
Offset: 0

Views

Author

Subhajit Bandyopadhyay, Dec 27 2023

Keywords

Comments

a_t(n), the phi-analog of the t-core partition function, is defined in the article by Gireesh, Ray and Shivashankar (2021). This sequence is for a_5(n).

Links

Crossrefs

Cf. A053723, A368495.

Programs

  • PARI
    q='q+O('q^71);
phi(q)=eta(q^2)^5 / (eta(q)*eta(q^4))^2;
gf=phi(-q^5)^5/phi(-q);
Vec(%) \\ Joerg Arndt, Dec 27 2023

Formula

G.f.: phi(-q^5)^5/phi(-q), where phi(q) is the Ramanujan's theta function phi (see A000122).

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